Editing Semi-continuity (section)

=== Upper and lower semicontinuity ===

A set-valued map <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is ''upper semicontinuous'' at <math>x \in \mathbb{R}^m</math> if for every open set <math>U \subset \mathbb{R}^n</math> such that <math>F(x) \subset U</math>, there exists a neighborhood <math>V</math> of <math>x</math> such that <math>F(V) \subset U.</math><ref name="freemanRobust"/>{{rp|Def. 2.1}}

A set-valued map <math>F: \mathbb{R}^m \rightrightarrows \mathbb{R}^n</math> is ''lower semicontinuous'' at <math>x \in \mathbb{R}^m</math> if for every open set <math>U \subset \mathbb{R}^n</math> such that <math>x \in F^{-1}(U),</math> there exists a neighborhood <math>V</math> of <math>x</math> such that <math>V \subset F^{-1}(U).</math><ref name="freemanRobust"/>{{rp|Def. 2.2}}

Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing <math>\mathbb{R}^m</math> and <math>\mathbb{R}^n</math> in the above definitions with arbitrary topological spaces.<ref name="freemanRobust"/>

Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty. 
An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map.<ref name="freemanRobust"/>{{rp|18}} 
For example, the function <math>f : \mathbb{R} \to \mathbb{R}</math> defined  by 
<math display=block>f(x) = \begin{cases}
-1   & \mbox{if } x < 0,\\
 1   & \mbox{if } x \geq 0
\end{cases}</math>
is upper semicontinuous in the single-valued sense but the set-valued map <math>x \mapsto F(x) := \{f(x)\}</math> is not upper semicontinuous in the set-valued sense. 
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Conversely, an upper semicontinuous set-valued function (in the set-valued sense) may not be upper semicontinuous in the single-valued sense sense it may not be single-valued.
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